L ECTURE 11: D ISCRETE -T IME D YNAMICAL S YSTEMS 2 I NSTRUCTOR : G - PowerPoint PPT Presentation

15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 11: D ISCRETE -T IME D YNAMICAL S YSTEMS 2 I NSTRUCTOR : G IANNI A. D I C ARO

O CCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠 # At 𝑠 ≈ 𝑠 # = 3.57 the map becomes § chaotic and the attractor changes from a finite to an infinite set of points The large window at 𝑠 ≈ 3.83 contains a § stable period-3 orbit 𝑔 𝑦 = 𝑠𝑦 1 − 𝑦 à the logistic map is 𝑦 012 = 𝑔(𝑦 0 ) § , 𝑦 017 = 𝑔(𝑔 𝑔 𝑦 0 ) = 𝑔 7 (𝑦 0 ) 𝑦 015 = 𝑔 𝑔 𝑦 0 § We are looking for 3-period cycles: every point 𝑞 in a 3-period cycle repeats § every 3 iterates à 𝑞 must satisfy 𝑞 = 𝑔 7 (𝑞) à 𝑞 is a fixed-point of the 𝑔 7 map § Unfortunately, the 𝑔 7 map is an 8-degree polynomial, a bit complex to study § 2

O CCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠 # 𝑠 = 3.835 , inside 3-period window § Intersections between the graph and the diagonal correspond to the solutions of 𝑔 7 𝑦 = 𝑦 § Only the black dots correspond to fixed points, and there are 3 of them, corresponding to the the 3-period cycle The slope of the function, |𝑔 : | is greater than 1 for the white dots, and § less than 1 for the black ones § For the other intersections, they correspond to fixed points or 1-period 3

O CCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠 # 𝑠 = 3.8 , before 3-period window 𝑠 = 3.835 , inside 3-period window § The 6 intersections of interest have vanished! § Not anymore periodic behavior For some 𝑠 between 3.8 and 3.835 the graph is tangent to the diagonal § At this critical value of 𝑠 , the stable and unstable 3-period cycles coalesce § and annihilate in a tangent bifurcation , that sets the beginning of the periodic window It can be computed analytically that this happens at 𝑠 = 1 + 6 § 4

O CCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠 # 𝑠 = 3.835 , inside 3-period window 𝑠 > 3.835 § Just after the tangent bifurcation, the slope at black dots (periodic points) is ≈ +1 (a bit less) For increasing values of 𝑠 , hills and valleys become steeper / deeper § The slope of 𝑔 7 at the black dots decreases steadily from ≈ +1 to -1. § When this occurs, a flip bifurcation happens, that causes each of the fixed periodic points to split in two § à the 3-period cycle becomes a 6-period cycle ! § … the process iterates, bringing the period doubling cascade! 5

W HAT ABOUT SENSITIVITY TO INITIAL CONDITIONS ? Chaos requires aperiodic orbits + sensitivity to initial conditions ü Aperiodic orbits arise § Sensitivity to initial conditions? Given an initial condition 𝑦 = , and an nearby point 𝑦 = + 𝜀 = , 𝜀 = ≈ 0 § 𝜀 expresses the separation between two near initial conditions § 𝜀 0 is the separation after 𝑜 iterates § If |𝜀 0 | ≈ |𝜀 = |𝑓 0B then 𝜇 is called the Lyapounov exponent (of 𝑦 = ) § § A positive Lyapounov exponent is a signature of chaos § Applies also to flows (like in the figure) 6

L YAPOUNOV EXPONENTS |𝜀 0 | ≈ |𝜀 = |𝑓 0B , taking the logarithm § 2 F G 𝜇 ≈ 0 ln| F H | § We can observe that 𝜀 0 = 𝑔 0 𝑦 = + 𝜀 = − 𝑔 0 (𝑦 = ) § I G J H 1F H KI G (J H ) 2 0 ln |(𝑔 0 ) : 𝑦 = | , for 𝜀 = → 0 2 𝜇 ≈ 0 ln| | = § F H 𝑔 0 is a function of all 𝑦 M , 𝑗 = 0, …,𝑜 à expansion by chain rule § 2 (𝑔 0 ) : 𝑦 M 2 ln | (𝑔 0 ) : 𝑦 M | 0K2 0 ∑ 0K2 0 ln| ∏ 𝜇 ≈ | = § MQ= MQ= If the limit for 𝑜 → ∞ exists, this is the Lyapounov exponent for the orbit § starting in 𝑦 = 𝜇 is the same for all points in the basin of attraction of an attractor § For stable fixed points and cycles, 𝝁 < 𝟏 § For chaotic attractors, 𝝁 > 𝟏 § 7

L YPUNOUV EXPONENTS FOR THE LOGISTIC MAP 8

A NOTHER M AP : H ENON MAP Classical settings for getting a chaotic behavior: 𝑏 = 1.4, 𝑐 = 0.3 Fractal dimension ≅ 1.26 9

F ROM N- DIMENSIONAL M APS TO C ELLULAR A UTOMATA § Let’s introduce topological notions and 2 2 , 𝑦 0 5 , … .𝑦 0 [ ) 𝑦 012 = 𝑔 2 (𝑦 0 constraints 5 2 , 𝑦 0 5 , …. 𝑦 0 [ ) 𝑦 012 = 𝑔 5 (𝑦 0 § Each variable represents the time-evolution ….. of a spatial location § Two variables can be (or not) neighbors [ 2 , 𝑦 0 5 , … .𝑦 0 [ ) 𝑦 012 = 𝑔 [ (𝑦 0 § Neighboring concepts go beyond metric § Generic k-dimensional map spaces …maybe too generic (complex!) § A variable’s evolution only depends on its neighbors à Influential variables § Not everybody is neighbor of everybody Cellular Automata (CA) § à Each map only depends on a § Mathematical simplification restricted set of neighbors, but the § Modeling spatial relations entire systems stays coupled § Dynamical model of many real-world systems 10

S TATE VARIABLES SPATIALLY BOUNDED ON L ATTICES : C ELLULAR A UTOMATA State variable ↔ State of Spatial location / Cell § Cell in a 1D, or 2D, or 3D Lattice § Example of selected neighborhood of 𝑦 M , represented by the set { 𝑦 MK2 , 𝑦 M12 } 𝑦 5 𝑦 7 𝑦 [ 𝑦 2 𝑦 MK2 𝑦 M 𝑦 M12 1D Example of selected neighborhood of 𝑦 [1M , represented by the set { 𝑦 [1MK2 , 𝑦 [1M12 , 𝑦 M , 𝑦 M12 ,𝑦 MK2 , 𝑦 5[1M , 𝑦 5[1M12 , 𝑦 5[1MK2 } 𝑦 5 𝑦 7 𝑦 2 𝑦 [ 𝑦 [12 𝑦 [15 𝑦 [17 𝑦 [1M 𝑦 5[ 2D 𝑦 [1] 11

CA S ARE L ATTICE MODELS Regular 𝒐 -dimensional discretization of a continuum § E.g., an 𝑜 -dimensional grid § § Periodic (toroidal) or non periodic structure § More abstract definition: Regular tiling of a space by a primitive cell Bethe lattice, ∞ -connected cycle-free 3D grid lattice, graph where each node is connected filled with spheres to 𝑨 neighbours, where 𝑨 is called of different colors the coordination number 12

C ELLULAR A UTOMATA Example of selected neighborhood of 𝑦 M , represented by the set { 𝑦 MK2 , 𝑦 M12 } 𝑦 5 𝑦 7 𝑦 [ 𝑦 2 𝑦 MK2 𝑦 M 𝑦 M12 1D 2 2 , 𝑦 0 5 ) 𝑦 012 = 𝑔 2 (𝑦 0 2 2 , 𝑦 0 5 , … .𝑦 0 [ ) 𝑦 012 = 𝑔 2 (𝑦 0 5 2 , 𝑦 0 5 , 𝑦 0 7 ) 𝑦 012 = 𝑔 5 (𝑦 0 5 2 , 𝑦 0 5 , …. 𝑦 0 [ ) 𝑦 012 = 𝑔 5 (𝑦 0 ….. ….. M , 𝑦 0 M MK2 , 𝑦 0 M12 ) 𝑦 012 = 𝑔 M (𝑦 0 [ 2 , 𝑦 0 5 , … .𝑦 0 [ ) ….. 𝑦 012 = 𝑔 [ (𝑦 0 [ [K2 , 𝑦 0 [ ) 𝑦 012 = 𝑔 [ (𝑦 0 Toroidal boundary conditions ….. 2 [ , 𝑦 0 2 , 𝑦 0 5 ) 𝑦 012 = 𝑔 2 (𝑦 0 2 [ , 𝑦 0 2 , 𝑦 0 5 ) 𝑦 012 = 𝑔(𝑦 0 ….. ….. Single map M , 𝑦 0 M , 𝑦 0 M MK2 , 𝑦 0 M12 ) 𝑦 012 = 𝑔 M (𝑦 0 M MK2 , 𝑦 0 M12 ) 𝑦 012 = 𝑔(𝑦 0 ….. ….. [ [K2 , 𝑦 0 [ , 𝑦 0 2 ) 𝑦 012 = 𝑔 [ (𝑦 0 [ [K2 , 𝑦 0 [ , 𝑦 0 2 ) 𝑦 012 = 𝑔(𝑦 0 13

CA S H ISTORICAL NOTES 14